It's about 37 miles from Moab to the 211 turnoff. On past trips, we've left Moab at 7:00 AM and are usually at or near the 211 turnoff about 8:00AM, give or take a little due to traffic or letting slower drivers catch up.
From the 211 turnoff, it's about another 31 miles to the trailhead but that's slower driving than on 191 plus we usually stop at the Newspaper Rock parking lot to see the pictographs and use the outhouse. We are usually airing down at the visitor's center parking lot about 9:00AM give or take, again, since some visit the last flushing toilet of the day. From the Visitor's Center, it's about another 10 minute drive to the trailhead.
My choice for where to park a car for the day would be the Visitor's Center parking lot. Even if you have to pay for both cars, I'll bet it's the safest.
1991 FJ80 converted to an HDT with an A442, Safari intercooler, 3" exhaust, factory Toyota subtank, 3" lift, 285's, Warn 9500xi, rear drawer system, 42qt Engel fridge. This rig is currently for sale